How to reparametrize by arc length?

**kandygirl16** · October 28th 2012, 01:40 PM

Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.

**TheEmptySet** · October 28th 2012, 02:28 PM

Originally Posted by kandygirl16

Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.

To find the length you need to integrate

$s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right)^2+\left( \frac{dy}{dt}\right)^2+\left( \frac{dz}{dt}\right)^2}}}, \quad t \in [a,b]$

This gives

$s=\int_{0}^{10}\sqrt{4\sin^2(t)\cos^2(t)+4\sin^2(t )\cos^2(t)+16t^2}dt}$

This function does not have a "nice" anti derivative.

Did you mean the helix $\mathbf{r}(t)=<\cos(t),\sin(t),2t>$ ?

**kandygirl16** · October 28th 2012, 02:58 PM

Hi, thanks for the reply. No unfortunately, this is the one I meant.

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